I began reading Paul Halmos' "Naive Set Theory", and encountered the "Axiom of Specification".
To every set $A$ and to every condition $S(x)$ there corresponds a set $B$ whose elements are exactly those elements $x$ of $A$ for which $S(x)$ holds.
Earlier in the same section, I learned that statements in set theory should be "sentences". A sentence was defined by
There are two basic types of sentences, namely, assertions of belonging, $x \in A$, and assertions of equality, $A = B$; all other sentences are obtained from such atomic sentences by repeated applications of the usual logical operators...
A more complete definition of a sentence follows, which can be read on Google Books here: http://goo.gl/XvK2B
I tried to translate the axioms and theorems in the book into sentences, but it seems like the Axiom of Specification is not a sentence. It refers to "every condition", but I have no way to build a sentence that refers to "every condition" because the atomic sentences only refer to sets.
Is the Axiom of Specification a sentence? If not, does that mean that statements about set theory do not need to be sentences? What other sorts of statements are allowed? (I'm using "statement" colloquially since I don't know the technical term.)